$\displaystyle \frac{1000}{\pi \left(\frac{500}{\pi}\right)^{2/3}} =$
$\displaystyle \frac{1000}{\pi} \cdot \frac{\pi^{2/3}}{500^{2/3}} =$
$\displaystyle \frac{2 \cdot 500}{\pi} \cdot \frac{\pi^{2/3}}{500^{2/3}} =$
$\displaystyle \frac{2 \cdot 500^{1/3}}{\pi^{1/3}} = 2\sqrt[3]{\frac{500}{\pi}}$